Let be an matrix over a field . Show that if , then is similar to a diagonal matrix whose diagonal entries are either 0 or 1.
(Compare to this previous exercise.)
Short Proof that I Thought Of Seconds after Publishing the Long Proof:
The minimal polynomial of divides . By Corollary 25 in D&F, is diagonalizable. Moreover, the elementary divisors of are all either or , so the diagonal entries of the Jordan canonical form of are either 0 or 1.
Long Proof that Missed the Point of the Whole Chapter but to which I am Too Attached to Delete Wholesale:
Let be such a matrix, and let be the Jordan canonical form of ; say . Now , so that is also idempotent. Note also that is upper triangular, with all entries above the first superdiagonal equal to 0. That is, if , then if or and if . (The diagonal entries of are unknown at this time.) Since , we have for each pair .
Fix . Now . That is, , so that is a root of . So (i.e. an arbitrary diagonal entry) is either 1 or 0.
Now fix and let . Now . Note that if , then we must have , since these entries are in the same Jordan block of . Now , and we have , a contradiction. Thus $latex for all appropriate . That is, is diagonal.